Modelling Fecal Cortisol Metabolites
Dr. Nicolas Ferry - Bavarian National Forest Park / Daniel Schlichting - StabLab
31 Jan 2025
assess red deer stress response towards hunting activities
– on 41 individual collared red deer
– within Bavarian Forest National Park
– using FCMs
FCMs: Faecal Cortisol Metabolites - a non-invasive method to measure stress through faecal samples
collared deer: red deer wearing a GPS-collar, which provides hourly location information
Euclidean Distance: Also known as \(L^2\) Distance. Reduces to Pythagorean Theorem for two Dimensions: \[d_{x,y} = ((x_1 - y_1)^2 + (x_2 - y_2)^2)^\frac{1}{2} \\ x,y \in \mathbb{R}^2 \]

model FCM levels on spatial and temporal distance to hunting activities
Expectation: FCM levels higher when closer in time and space
Movement: contains the location and datetime1 of the 41 collared deer in the period Feb 2020 - Feb 2023. In total approx. 740 000 observations2
Hunting Events: contains location and date of hunting events in the National Park - in total 1270 events, 890 of them with full timestamp
FCM Stress: contains information of 809 faecal samples, including:
– the location of the sample
– the time of sampling
– the DNA-matched collared deer
– the time when the deer was at the location
Reproduction Success: observations of 16 collared deer on:
– if they were pregnant in one year
– if they were accompanied by a calf in one year
Note: The defecation location is not the deer’s location at the time of the stress event.
TBD: Illustration.
We introduce 4 Parameters:
A hunting event is considered relevant to an FCM sample, if
Among the relevant hunting events, the most relevant one is defined by the proximity criterion:
The scoring function is defined as TBD.
We suggest eight different Datasets for Modelling
For Modelling, we consider the following covariates:
| Model | Type | Non-Parametric Effects | Linear Effects | Random Intercept | Distribution Assumption |
|---|---|---|---|---|---|
| A | GAM | Time Difference, Distance, Sample Delay, Day of Year | Pregnant, Number Other Hunts | None | Gaussian |
| B | GAM | Time Difference, Distance, Sample Delay, Day of Year | Pregnant, Number Other Hunts | None | Gamma |
| C | GAMM | Time Difference, Distance, Sample Delay, Day of Year | Pregnant, Number Other Hunts | Deer | Gaussian |
| D | GAMM | Time Difference, Distance, Sample Delay, Day of Year | Pregnant, Number Other Hunts | Deer | Gamma |
\(FCM_i \sim \mathcal{N}(\mu_i, \sigma^2)\)
Identity Link: \(E(FCM_i) = \mu_i = \eta_i\)
Linear Predictor: \[ \begin{equation} \begin{gathered} \eta_i = \beta_0 + \beta_1\,Pregnant_i +\\ \beta_2\,Number\,Other\,Hunts_i + f_1(Time\,Diff_i) + \\ f_2(Distance_i) + f_3(Sample\,Delay_i) + f_4(Day\,of\,Year_i) \end{gathered} \end{equation} \]










\(FCM_i \sim \mathcal{Ga}(\nu, \frac{\nu}{\mu_i})\)
For better Interpretability we use the Log-Link: \(E(FCM_i) = \mu_i = exp(\eta_i)\)
Linear Predictor: \[ \begin{equation} \begin{gathered} \eta_i = \beta_0 + \beta_1\,Pregnant_i +\\ \beta_2\,Number\,Other\,Hunts_i + f_1(Time\,Diff_i) + \\ f_2(Distance_i) + f_3(Sample\,Delay_i) + f_4(Day\,of\,Year_i) \end{gathered} \end{equation} \]










\(FCM_{i\,j} \sim \mathcal{N}(\mu_{i\,j}, \sigma^2)\)
Identity Link: \(E(FCM_{i\,j}) = \mu_{i\,j} = \eta_{i\,j}\)
Linear Predictor: \[ \begin{equation} \begin{gathered} \eta_{i\,j} = \beta_0 + \beta_1\,Pregnant_{i\,j} +\\ \beta_2\,Number\,Other\,Hunts_{i\,j} + f_1(Time\,Diff_{i\,j}) + \\ f_2(Distance_{i\,j}) + f_3(Sample\,Delay_{i\,j}) + f_4(Day\,of\,Year_{i\,j}) \end{gathered} \end{equation} \] with: \(\gamma_j \overset{\mathrm{iid}}{\sim} \mathcal{N}(0, \sigma_{\gamma}^2)\)










Gamma as distributional assumption, as FCM level is positive.
Log link for interpretability.
Let \(i = 1,\dots,N\) be the indices of deer and \(j = 1,\dots,n_i\) be the indices of FCM measurements for each deer.
\[ \begin{eqnarray} \textup{FCM}_{ij} &\sim& \mathcal{Ga}\left( \nu, \frac{\nu}{\mu_{ij}} \right) \\ \mu_{ij} &=& \mathbb{E}(\textup{FCM}_{ij}) = \exp(\eta_{ij}) \\ \eta_{ij} &=& \beta_0 + \beta_1 \textup{Pregnant}_{ij} + \beta_2 \textup{NumberOtherHunts}_{ij} + \\ && f_1(\textup{TimeDiff}_{ij}) + f_2(\textup{Distance}_{ij}) + f_3(\textup{SampleDelay}_{ij}) + f_4(\textup{DefecationDay}_{ij}) + \\ && \gamma_{i}, \\ \gamma_i &\sim& \mathcal{N}(0, \sigma_\gamma^2). \end{eqnarray} \]










Not many observations after datafusion left for robust modelling
Trade-off between spatial and temporal distance
Sample Delay seems to be significant
Modelling Outcomes don’t show much difference
Trade-off between Complexity and Explainability
How to minimize spatial and temporal distance at the same time?
How to use a bigger Part of the Data?
Effect of Hunting on Red Deer